Interaction of Global-scale Atmospheric Vortices: Modeling based on Hamiltonian Dynamic System of Antipodal Point Vortices on Rotating Sphere

Abstract

We get point vortices dynamics equations on a rotating sphere surface directly from the hydrodynamic equations as representing their weak exact solution contrary to the conventional case of the use of a kinematic relationship between a given singular vortex field and velocity field. It is first time that the effect of a sphere rotation on the vortices interaction is accounted for exactly. We prove that only antipodal vortices (APV) pair with two point vortices in the diameter-conjugated points of a sphere with equal by quantity but different sign circulations, may be correctly considered in the frame of hydrodynamics equations as an elementary (stationary, not self-affecting) singular point object on a sphere. We suggest using the axis connecting the two point vortices in an APV for describing of an axis of rotation of the global vortices introduced in (Barrett, 1958) to reflect the observed global rotation of atmospheric to the west. On the base of corresponding exact solutions, we show acceptability of modelling of the stable blocks of splitting flow type when in line with the exact accounting for the sphere rotation, we define also conditions of the polar vortices affecting on the stability boundaries of the block modes.

Point vortices dynamics on a rotating sphere and modeling of global atmospheric vortices interaction

“…According to a long-term observations (see Appendix C in 31 ) the cyclonic-anticyclonic vortex pair of the ACAs over North Atlantic, usually only slightly deviates from the equilibrium annualmean coordinates that corresponds to the case illustrated by Fig. 1b.…”

“…According to (3.1), each Lagrangian particle preserves the value of absolute vorticity in a thin layer of liquid on a rotating sphere. Therefore, as in 25…”

“…Until now, we have not taken into account the permanently existing polar vortices which may affect substantially stability of the equilibrium state of the above mentioned paired ACAs even in a linear approximation 31 .…”

“…A simple non-trivial zero solution of the absolute vorticity conservation equation is used for the exact describing APV dynamics on the rotating sphere, when in contrast to 21 it is possible to realize condition 0   . A zero value of absolute vorticity corresponds to a non-zero stream function on the sphere, which influenced on 2N-dimensional dynamical system of equations for the APVs on the rotating sphere 25 23 . Due to the zero value of absolute vorticity regular field the well-known 21 19 , 10 , 6 , 3 …”